Mathematical Sciences: Representations of Three-Manifold Groups

小行星9505053 卡森将研究几何猜想三维流形，提出了瑟斯顿在1982年。 这个猜想的证明将完成闭3-流形的拓扑分类，提供对3-流形拓扑性质的洞察，并导致有效的方法来确定两个给定的3-流形是否拓扑等价。 卡森将探索瑟斯顿猜想的一种非常直接的方法;给定3-流形M的拓扑描述，他将尝试找到与M上的几何结构密切相关的M的三角剖分。 柯比将继续他的工作（经常与保罗梅尔文）在拓扑量子场论。 他将试图解释孔采维奇和瓦西里耶夫不变量，使用某些余维2流形的交叉点在紧致化的产品与对角线删除给定的3流形。 他还将继续研究公式的维滕-Reshetikhin-Turaev不变量，寻找拓扑的解释和应用。 由于我们生活的空间是三维的，而我们一生中只是通过日常事务来完善我们的三维几何直觉，因此，数学家往往更有能力回答关于高维几何对象的问题，而不是关于三维中的类似对象的问题，这总是让外行感到惊讶。 然而，情况确实如此。 拓扑学中最著名的猜想是世纪之交提出的三维庞加莱猜想。 另一方面，它的高维类似物已经全部解决。 该项目的研究直接针对这些众所周知的棘手问题。 ***

9505053 Casson Casson will study the Geometrization Conjecture for 3- dimensional manifolds, proposed by Thurston in 1982. A proof of this conjecture would complete the topological classification of closed 3-manifolds, provide insight into topological properties of 3-manifolds, and lead to efficient methods to determine whether two given 3-manifolds are topologically equivalent. Casson will explore a very direct approach to Thurston's conjecture; given a topological description of a 3-manifold M, he will attempt to find triangulations of M that are closely related to a geometric structure on M. Kirby will continue his work (often with Paul Melvin) in Topological Quantum Field Theory. He will try to explain the Kontsevich and Vassiliev invariants, using intersections of certain codimension-two manifolds in a compactification of products of the given 3-manifold with diagonals deleted. He will also continue to study the formulae for the Witten-Reshetikhin-Turaev invariants, looking for topological interpretations and applications. Since the space that we live in is three-dimensional, and all our lives we perfect our three-dimensional geometric intuition simply by going about our everyday business, it is always surprising to the layman to learn that mathematicians' are frequently more capable of answering questions about higher dimensional geometric objects than about the analogous objects in dimension three. This is, however, very much the case. The most famous outstanding conjecture in topology is the three-dimensional Poincare conjecture, made at the turn of the century. On the other hand, its higher dimensional analogues have all been settled. The research of this project is directly addressed to these notoriously recalcitrant problems. ***